Splitting Fields

Summary. In this article we further develop field theory in Mizar [1], [2]: we prove existence and uniqueness of splitting fields. We define the splitting field of a polynomial p ∈ F [X] as the smallest field extension of F, in which p splits into linear factors. From this follows, that for a splitting field E of p we have E = F (A) where A is the set of p’s roots. Splitting fields are unique, however, only up to isomorphisms; to be more precise up to F -isomorphims i.e. isomorphisms i with i|F = Id F . We prove that two splitting fields of p ∈ F [X] are F -isomorphic using the well-known technique [4], [3] of extending isomorphisms from F 1 → F 2 to F 1(a) → F 2(b) for a and b being algebraic over F 1 and F 2, respectively.

SIMETRISASI BENTUK KANONIK JORDAN

If the characteristic polynomial of a linear operator is completely factored in scalar field of then Jordan canonical form of can be converted to its rational canonical form of , and vice versa. If the characteristic polynomial of linear operator is not completely factored in the scalar field of ,then the rational canonical form of can still be obtained but not its Jordan canonical form matrix . In this case, the rational canonical form of can be converted to its Jordan canonical form by extending the scalar field of to Splitting Field of minimal polynomial of , thus forming the Jordan canonical form of over Splitting Field of . Conversely, converting the Jordan canonical form of over Splitting Field of to its rational canonical form uses symmetrization on the Jordan decomposition basis of so as to form a cyclic decomposition basis of which is then used to form the rational canonical matrix of

Ring and Field Adjunctions, Algebraic Elements and Minimal Polynomials

Summary In [6], [7] we presented a formalization of Kronecker’s construction of a field extension of a field F in which a given polynomial p ∈ F [X]\F has a root [4], [5], [3]. As a consequence for every field F and every polynomial there exists a field extension E of F in which p splits into linear factors. It is well-known that one gets the smallest such field extension – the splitting field of p – by adjoining the roots of p to F. In this article we start the Mizar formalization [1], [2] towards splitting fields: we define ring and field adjunctions, algebraic elements and minimal polynomials and prove a number of facts necessary to develop the theory of splitting fields, in particular that for an algebraic element a over F a basis of the vector space F (a) over F is given by a 0 , . . ., an− 1, where n is the degree of the minimal polynomial of a over F .

Quadratic Forms

This chapter presents a few standard definitions and results about quadratic forms and polar spaces. It begins by defining a quadratic module and a quadratic space and proceeds by discussing a hyperbolic quadratic module and a hyperbolic quadratic space. A quadratic module is hyperbolic if it can be written as the orthogonal sum of finitely many hyperbolic planes. Hyperbolic quadratic modules are strictly non-singular and free of even rank and they remain hyperbolic under arbitrary scalar extensions. A hyperbolic quadratic space is a quadratic space that is hyperbolic as a quadratic module. The chapter also considers a split quadratic space and a round quadratic space, along with the splitting extension and splitting field of of a quadratic space.

Residually Pseudo-Split Buildings

This chapter presents results about a residually pseudo-split Bruhat-Tits building Ξ‎L. It begins with a case for some quadratic space of type E⁶, E₇, and E₈ in order to identify an unramified extension such that the residue field is a pseudo-splitting field. It then considers a wild quaternion or octonion division algebra and the existence of an unramified quadratic extension L/K such that L is a splitting field of the quaternion division algebra. It also discusses the properties of an unramified extension L/K and shows that every exceptional Bruhat-Tits building is the fixed point building of a strictly semi-linear descent group of a residually pseudo-split building.

Reductions of one-dimensional tori

Consider a non-split one-dimensional torus defined over a number field [Formula: see text]. For a finitely generated group [Formula: see text] of rational points and for a prime number [Formula: see text], we investigate for how many primes [Formula: see text] of [Formula: see text] the size of the reduction of [Formula: see text] modulo [Formula: see text] is coprime to [Formula: see text]. We provide closed formulas for the corresponding Dirichlet density in terms of finitely many computable parameters. To achieve this, we determine in general which torsion fields and Kummer extensions contain the splitting field.